Systems and methods for compressing plasma

ABSTRACT

Embodiments of systems and methods for compressing plasma are described in which plasma pressures above the breaking point of solid material can be achieved by injecting a plasma into a funnel of liquid metal in which the plasma is compressed and/or heated.

CROSS-REFERENCE TO RELATED APPLICATIONS

This application is a continuation of U.S. patent application Ser. No.12/699,725, filed Feb. 3, 2010, entitled “SYSTEMS AND METHODS FORCOMPRESSING PLASMA,” now U.S. Pat. No. 8,537,958, which claims thebenefit under 35 U.S.C. §119(e) of U.S. Provisional patent applicationNo. 61/149,886, filed Feb. 4, 2009, entitled “SYSTEMS AND METHODS FORACCELERATING AND COMPRESSING A PLASMA;” each of the foregoing is herebyincorporated by reference herein in its entirety.

BACKGROUND

1. Field

The present disclosure relates to embodiments of systems and methods forcompressing plasma. In certain such embodiments, a plasma toroid iscompressed using a liquid metal funnel.

2. Description of the Related Art

Various systems for heating and compressing plasmas to high temperaturesand densities have been described. One approach for accomplishing plasmaheating and compression by spherical focusing of a large amplitudeacoustic pressure wave in a liquid medium is described in U.S. PatentPublication No. 2006/0198486, published Sep. 7, 2006, entitled “PressureWave Generator and Controller for Generating a Pressure Wave in a FusionReactor”, which is hereby incorporated by reference herein in itsentirety. In certain embodiments of this approach, a plurality ofpistons is arranged around a substantially spherical vessel containing aliquid medium. A vortex or cavity is created in the liquid medium. Thepistons are accelerated and strike the outer wall of the vesselgenerating an acoustic wave. The acoustic wave generated in the liquidmedium converges and envelopes a plasma that is introduced into thevortex, thereby heating and compressing the plasma.

A pressure wave generator of the type described in U.S. PatentPublication No. 2006/0198486 can be employed in a Magnetized TargetFusion (MTF) reactor as described, for example, in U.S. PatentPublication No. 2006/0198483, published Sep. 7, 2006, entitled“Magnetized Plasma Fusion Reactor,” which is hereby incorporated byreference herein in its entirety. In certain such implementations, amagnetized plasma is introduced into a vortex that is created in theliquid medium, such as molten lead-lithium (PbLi). The acoustic waveproduced by the impact of pistons surrounding the spherical reactorvessel can compress the magnetized plasma to high density andtemperature.

In some embodiments of the above-described devices, compressed gas suchas steam or air can be used to accelerate the pistons. Typically thedesired piston impact velocity for plasma compression is of the order of100 m/s, and so generally a compressed gas pressure of about 1,300 psiis used to accelerate the pistons. To achieve the symmetry of implosionthat may be useful or desirable in some implementations, the timing ofthe piston firing, trajectory, and impact is precisely controlled foreach piston. For example, for some plasma compression implementations,all the pistons preferably strike the vessel wall within about 1 μs ofeach other. In some such implementations, a servo control system can beused to measure precisely the position of each piston and control itstrajectory to attain the requisite impact time.

Whilst certain embodiments of such mechanical compression systems areattractive from, for example, a cost perspective, certain suchimplementations may need frequent maintenance, especially inapplications where the repetition frequency of piston firing is high.

SUMMARY

Embodiments of systems and methods for compressing plasma are disclosed.Some embodiments comprise electrically accelerating a plasma, forexample, by using a plasma accelerator such as, e.g., a rail gun. Theplasma can be accelerated into a funnel of liquid metal where the plasmais further compressed. The use of the liquid metal allows high plasmadensities to be achieved, because, in some embodiments, the pressureattained can be higher than the breakpoint or yield strength of solidmaterials typically used in the apparatus itself.

In certain embodiments, a low density and temperature spheromak ortoroidal plasma is formed using a plasma gun, for example, a magnetizedcoaxial gun. The toroidal plasma is electrically accelerated, compressedand heated to a high density and temperature using a plasma accelerator(e.g., a tapered rail gun) that extends towards a liquid metal funnel.The liquid metal funnel in some implementations can be formed of moltenmetal such as, for example, molten lead-lithium (PbLi). In variousembodiments, the toroidal plasma can be formed as a field-reversedconfiguration (FRC) or other compact toroid.

In some implementations, the plasma can include a fusionable materialsuch as, for example, isotopes of light elements (e.g., deuterium,tritium, helium-3, lithium-6, and/or lithium-7). The higher plasmadensities and/or temperatures that are achievable in some suchimplementations can be sufficient for the initiation of fusionreactions. Some fusion reactions produce neutrons. Therefore, someembodiments of the system can be configured as neutron sources. Someembodiments of the systems and methods may provide sufficient fusionreactions for net energy production to occur (e.g., above breakeven).

An embodiment of an apparatus for compressing plasma is disclosed. Theapparatus comprises a plasma gun configured to generate a compact toroidof plasma, a plasma accelerator, and a liquid funnel system. The plasmaaccelerator has a first end, a second end, and a longitudinal axisbetween the first end and the second end. The plasma accelerator isconfigured to receive the compact toroid at the first end and toaccelerate the compact toroid along the longitudinal axis toward thesecond end. The liquid funnel system comprises a liquid funnel having asubstantially cylindrical passage substantially aligned with thelongitudinal axis of the plasma accelerator. The passage has a firstinner diameter at a top end of the passage and a second inner diameterat a bottom end of the passage. The second inner diameter can be lessthan the first inner diameter in some embodiments. The liquid funnelsystem is configured to receive the compact toroid from the second endof the plasma accelerator and to compress the compact toroid as thecompact toroid moves along the passage from the top end toward thebottom end. The system can be configured such that a pressure of thecompact toroid when below the top end is greater than a pressure of thecompact toroid when above the top end.

An embodiment of a liquid metal funnel system configured for compressingplasma is disclosed. The liquid metal funnel system comprises a liquidmetal funnel having a substantially cylindrical passage having a firstinner diameter at a first end of the passage and a second inner diameterat a second end of the passage. The second inner diameter can be lessthan the first inner diameter. The liquid metal funnel can be orientedsuch that the first end of the passage is higher than the second end ofthe passage. The liquid metal funnel can be configured to receive aplasma from a plasma injector and to compress the plasma as the plasmamoves along the passage from the first end toward the second end.

An embodiment of a method of compressing a plasma is disclosed. Themethod comprises generating a toroidal plasma, accelerating the toroidalplasma along a longitudinal direction, and introducing the acceleratedtoroidal plasma into a passage in a liquid funnel. The passage can havea first size at a first end of the passage and a second size at a secondend of the passage. The second size can be smaller than the first size.The method can also include compressing the toroidal plasma as thetoroidal plasma moves from the first end of the passage toward thesecond end of the passage.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1A is a schematic cross-sectional diagram showing an embodiment ofa system compressing a plasma in a tapered liquid metal funnel. In thisembodiment, a plasma gun forms a compact toroid that is accelerated by aplasma accelerator toward the liquid metal funnel.

FIG. 1B is a schematic cross-sectional diagram showing anotherembodiment of a system for compressing a plasma in a liquid metal funnelsystem. In this embodiment, the funnel system comprises a liquid metalfunnel and an axial liquid metal guide disposed substantially along acentral axis of the funnel. In this embodiment, the plasma acceleratorcomprises a plasma restrictor that comprises a constriction in apropagation channel of the accelerator.

FIG. 1C is a schematic cross-sectional diagram showing anotherembodiment of a system for compressing a plasma in a liquid metal funnelsystem. In this embodiment, the plasma accelerator comprises a plasmarestrictor that comprises one or more magnetic coils.

FIG. 1D is a perspective cutaway view of an embodiment of a system forcompressing a plasma. The embodiment shown in FIG. 1C is generallysimilar to the embodiment schematically shown in FIG. 1B.

FIG. 2 is a graph indicating an example calculation of the energy of theplasma to achieve the Lawson criteria for various plasma densities andan example calculation of the magnetic pressure of a plasma at variousplasma densities. These example calculations are based on Bohm diffusionand certain other assumptions described below. Note that a pressure of 1atmosphere (atm) is about 10⁵ Pa.

FIG. 3 is a schematic cross-sectional diagram showing an example of atoroidal plasma within a tapered liquid metal funnel.

FIG. 4 is a graph showing an example calculation of the plasma energy toachieve the Lawson criteria for various plasma densities, taking intoaccount various power losses that can occur in an example embodiment ofa plasma compression system.

DETAILED DESCRIPTION OF EMBODIMENTS

Tapered coaxial plasma spheromak accelerators have been built andstudied in the past for, e.g., x-ray production, tokomak fuelling, andplasma physics research. However, the maximum achievable magneticpressure has been limited by the strength of the solid materials used inthe apparatus (e.g., a fracture limit, yield strength, or breakpoint ofthe solid materials). In certain embodiments of the present approach,the magnetic pressure that can be achieved has been increasedsignificantly beyond this limit by using a tapered or funnel-shapedliquid metal tube as described in more detail below.

With reference to the drawings, FIGS. 1A-1D schematically illustrateseveral embodiments of a system 1000 that can be used to accelerate andcompress a plasma. The embodiments shown in FIGS. 1A-1D comprise aplasma gun 100 configured to generate a toroidal plasma (for example, acompact toroid such as, e.g., a spheromak), a plasma accelerator 110configured to accelerate the plasma along a longitudinal axis 115 of theaccelerator 110, and a liquid metal funnel system 120 into which theplasma accelerated by the accelerator 110 is introduced for furthercompression. In various embodiments, the plasma gun 100 may comprise amagnetized plasma gun that has a gun axis that is substantially alignedor coaxial with the longitudinal axis 115 of the accelerator 110. Insome embodiments, the plasma gun 100 comprises a Marshall-type plasmagun. In various embodiments, the plasma accelerator 110 may comprise arail gun configured to accelerate the plasma using magnetic and/orelectromagnetic forces. In some embodiments, the plasma accelerator 110can provide some degree of plasma compression as the plasma isaccelerated along the longitudinal axis 115. For example, the rail guncan comprise one or more tapered electrodes to compress the plasmaduring acceleration along the longitudinal axis 115. The liquid metalfunnel system 120 may comprise a liquid metal funnel, cylinder, or tube8 having a passage substantially aligned with the longitudinal axis 115of the accelerator 110. In some embodiments, a cross-section and/orinner diameter of the passage can change from an upper end of the funnelto a lower end of the funnel, e.g., the cross-section (and/or innerdiameter) can decrease to allow the plasma to be compressed as theplasma moves below the upper end and toward the lower end. In certainembodiments, the plasma gun 100 and/or the plasma accelerator 110 arepositioned substantially above the liquid funnel system 120. In certainembodiments, the upper end of the funnel 8 is substantially above thelower end of the funnel 8.

The toroidal plasma generated by the plasma gun 100 can be a compacttoroid such as, e.g., a spheromak, which is a toroidal plasma confinedby its own magnetic field produced by current flowing in the conductiveplasma. In other embodiments, the compact toroid can be a field-reversedconfiguration (FRC) of plasma, which may have substantially closedmagnetic field lines with little or no central penetration of the fieldlines.

As schematically illustrated in the embodiments shown in FIGS. 1A-1D,gas from one or more tanks 4 is introduced into the gun by fast puffvalves 3. In some implementations, the initial gas pressure is about 15pounds per square inch (psi) (e.g., about 1.03×10⁵ Pa). The gas maycomprise a fusionable material. For example, the fusionable material maycomprise one or more isotopes of light elements such as, e.g., isotopesof hydrogen (e.g., deuterium and/or tritium), isotopes of helium (e.g.,helium-3), and/or isotopes of lithium (e.g., lithium-6 and/orlithium-7). Other fusionable materials can be used. Combinations ofelements and isotopes can be used. For example, in some implementations,a 50% deuterium-50% tritium gas mixture is introduced from the tank 4using about 100 puff valves 3. Each pulse from the valves introducesabout 2 mg of gas, in one implementation. In other embodiments, adifferent number of valves can be used and/or a different mass of gascan be introduced. In other implementations, the percentages ofdeuterium and tritium, respectively, can be different from 50%-50%.

Coils 5 induce a magnetic field in the space between an outer electrode7 and a formation electrode 14. The coils 5 can be configured to providea mostly radial stuffing magnetic field of about 0.8 Tesla in someimplementations. In the embodiment schematically illustrated in FIGS.1A-1C, the formation electrode 14 is substantially cylindrical, and theouter electrode 7 is tapered inward toward the liquid metal funnelsystem 120. In the embodiments schematically illustrated in FIGS. 1B-1D,the magnetic field is produced by 3 magnetic coils 5 a, 5 b, and 5 c,although fewer or greater numbers of coils may be used in otherembodiments. In some embodiments, the coil 5 a comprises about 140 turnsof hollow square 6 mm×6 mm copper wire. During operation of the system,the wire can carry a current of about 1000 Amps at a voltage of about630 V dissipating about 630 kW. Coil 5 b comprises about 224 turns ofhollow square 6 mm×6 mm copper wire carrying a current of about 1000Amps at a voltage of about 832 V dissipating about 830 kW. Coil 5 ccomprises about 552 turns of hollow square 6 mm×6 mm copper wirecarrying a current of about 1000 Amps at a voltage of about 844 Vdissipating about 840 kW. In some embodiments, the coils 5 a, 5 b, and 5c will run substantially continuously during operation of the system. Insome embodiments, a cooling system (not shown) provides water (oranother coolant), which flows in the hollow wires to cool them.

In certain implementations of the system, it is desirable to introducegas only between the outer electrode 7 and the formation electrode 14.In certain such implementations, the valves 3 open and closesufficiently rapidly to introduce the gas so that it is substantiallyconfined between the electrodes 7 and 14. For example, at roomtemperature (e.g., about 20 C), the thermal velocity of the gas is about900 m/s. If, for example, the distance between the electrodes 7 and 14is about one meter, the gas could be injected for a duration of lessthan about 1 ms to provide gas for the generation of each compacttoroid. In some implementations, Parker Series 99 valves can be used(available from Parker Hannifin, Cleveland, Ohio).

In some embodiments, the formation electrode 14 is electricallyconnected to a capacitor bank 1. In some such embodiments, the capacitorbank 1 can comprise a capacitance of about 4.1 mF and the bank can becharged at a voltage of about 22 kV. In some cases, the capacitor bank 1comprises about eighty 52 μF individual capacitors (e.g., GeneralAtomics Energy Products (San Diego, Calif.), model 33677 capacitors).The individual capacitors can be connected in parallel. The capacitorbank 1 can be connected to the formation electrode 14 using atransmission line. In some embodiments, the total inductance of thetransmission line and capacitors is about 20 nF, which advantageouslyprovides a sufficiently fast electric discharge.

During operation of the system 1000, when the gas introduced by the puffvalves 3 achieves a suitable pressure between the electrodes 7 and 14,the capacitor bank 1 discharges in the gas, turning the gas into aplasma. The discharge can occur when the capacitor bank voltage exceedsthe breakdown voltage of the gas (which can depend on the gas pressure).In some implementations, the bank 1 discharges when the gas pressure isabout 10 mTorr (e.g., about 1.3 Pa). The discharge can occur at othergas pressures in other embodiments. In the embodiment shown in FIG. 1A,a switch 2 is activated to discharge the capacitor bank 1 through thegas, generating a plasma. A possible advantage of embodiments using theswitch 2 is that the switch can be activated so that the dischargeoccurs when the gas is at a desired pressure, which may allow increasedflexibility during operation.

The current rises (e.g., to about 3 MAmp in about 20 μs in some cases),and the magnetic field from this current forces the plasma in thedownward direction in FIGS. 1A-1C, toward the plasma accelerator 110.The stuffing magnetic flux from the coils 5 wraps itself around theplasma. The magnetic field reconnects to form closed magnetic surfaces,and the plasma forms a compact toroid. For example, the toroid may be aspheromak 16 having a relatively low density (e.g., about 10¹⁵ cm⁻³, insome cases) and temperature (e.g., about 20 eV, in some cases).

In some implementations of the system 1000, after a relatively smalldelay (e.g., about 30 μs, in some cases) to allow the magnetic fields toreconnect and/or to allow turbulence, if present, to settle, thespheromak 16 is accelerated and compressed by the plasma accelerator 110toward the liquid metal funnel system 120.

For example, in the embodiments schematically illustrated in FIGS.1A-1D, acceleration electrodes 6 are connected to a second capacitorbank 11, which is used to provide energy to the plasma. In someembodiments, the capacitor bank 11 has a capacitance of about 2.6 mF andis charged at a voltage of about 88 kV. In some such embodiments, thecapacitor bank 11 comprises about 100 pairs of 52 μF, 44 kV individualcapacitors with each pair electrically connected in series, and the 100pairs electrically connected in parallel. In some embodiments, thecapacitors comprise General Atomics Energy Products (San Diego, Calif.),model 32283 capacitors. In some implementations, the capacitor bank 11is electrically connected to the acceleration electrode 6 using asubstantially disk-shaped transmission line 15 substantially surroundingthe coaxial gun 100 to reduce or minimize inductance. The currentrise-time of some embodiments is about 40 μs because of the relativelylarge capacitance of the bank 11. In some embodiments, a substantiallydisk-shaped transmission line can be used to electrically connect thecapacitor bank 1 to the formation electrode 14. In some embodiments, twosubstantially disk-shaped transmission lines are used: a firstsubstantially disk-shaped transmission line electrically connecting thecapacitor bank 1 to the formation electrode, and a second substantiallydisk-shaped transmission line electrically connecting the capacitor bank11 to the acceleration electrode 6.

The plasma accelerator 110 comprises a plasma propagation channel 114through or along which the toroidal plasma 16 is accelerated. Forexample, as schematically illustrated in FIGS. 1A-1D, the accelerationelectrode 6 can be disposed within the outer electrode 7, and the plasmapropagation channel 114 comprises space between the electrodes 6 and 7.The plasma propagation channel 114 can have a cross-section(perpendicular to the longitudinal axis 115) that changes (in shape,size, width, spacing, and/or any other way) from a first end 112 a to asecond end 112 b of the accelerator. For example, in the embodimentsillustrated in FIGS. 1A-1D, at least one of the electrodes 6 and 7 canbe tapered from the first end 112 a of the accelerator 100 (e.g., nearthe plasma gun 100) to the second end 112 b of the accelerator 100(e.g., near the funnel system 120). For example, in some embodiments,the radius of the accelerator 110 (e.g., a radius from the longitudinalaxis 115 to the center of the channel 114) decreases by a factor ofabout 30 from the first end 112 a to the second end 112 b. In otherembodiments, the radius of the accelerator 110 decreases from the firstend to the second end by a factor of about 2, about 5, about 10, about20, about 50, about 100, or some other factor. In various embodiments,the radius decrease of the accelerator from the first end to the secondend can be in a range from about 10 to about 50, in a range from about20 to about 40, or some other range.

With further reference to the embodiments schematically illustrated inFIGS. 1A-1D, the magnetic force of the plasma accelerator 110accelerates the toroidal plasma 16 between the tapered coaxialelectrodes 6 and 7 and heats and compresses the plasma to highertemperature and density, forming a compressed toroidal plasma 12.

The configuration of the electrodes 6, 7 can be selected to provide adesired amount of compression as the plasma moves from the first end 112a to the second end 112 b of the accelerator 110. For example, one ormore factors including the tapering, shape, and/or spacing of theelectrodes 6, 7 can be selected to provide a desired compression. In thecase of some toroidal configurations of plasma (e.g., compact toroids),the compression of the plasma in some implementations of the system 1000can be measured in terms of a radial compression of the toroid (e.g., aratio of the radius of the toroid when in a first position in the systemto the radius of the toroid when in a second position in the system).For example, in some embodiments, the radial compression of the plasmaas the plasma moves from the first end 112 a to the second end 112 b ofthe accelerator 110 is about 30:1. The radial compression of the plasmain the accelerator 110 can be different in other embodiments such as,for example, about 2:1, about 5:1, about 10:1, about 15:1, about 20:1,about 30:1, about 50:1, about 100:1, etc. In various embodiments, thecompression of the plasma in the accelerator 110 can be in a range fromabout 10:1 to about 50:1, in a range from about 20:1 to about 40:1, orin some other appropriate range. In some embodiments, tapering of theelectrodes 6, 7 is not used in the accelerator 110, and there issubstantially no compression of the plasma in the accelerator 110.

In other embodiments, the plasma accelerator 110 may be configured sothat the outer electrode 7 acts as the acceleration electrode. In otherembodiments, both electrodes 6 and 7 can be used to electromagneticallyaccelerate the plasma from the first end to the second end. In otherembodiments, additional electrodes can be used (e.g., to assiststabilizing the plasma and/or to inhibit tilting of the toroid in thechannel 114).

The electrodes 6, 7, and/or 14 can be formed from electricallyconductive metal. The electrodes 6, 7, and/or 14 can be formed in one ormore sections. For example, in some embodiments, the electrodes 6, 7,and/or 14 comprise one or more stainless steel 304 plates or sheetshaving a thickness of about 5 mm. The sections of the electrodes can bejoined together by welding, fasteners (e.g., bolts), etc. In otherembodiments, the electrodes can be formed from additional and/ordifferent materials and/or thicknesses of material. In someimplementations, the plasma can become sufficiently hot to at leastpartially vaporize some of the electrodes. Vaporization of the electrodemay in some cases contaminate the plasma with metallic impurities thatcan cool down the plasma. Therefore, in certain implementations thatutilize electrodes that can (at least partially) vaporize, one or moreof the electrodes 6, 7, and 14 can be coated with a high melting pointmaterial such as, e.g., tungsten. The coating material can be selectedso that the melting point of the coating material (e.g., tungsten) isgreater than the melting point of the electrode material (e.g.,stainless steel). For example, in some implementations, tungsten isplasma sprayed on the electrode material (e.g., stainless steel 304).For example, Flamespray Northwest in Seattle, Wash., provides plasmaspraying services. In other embodiments, the high melting point materialcan be layered or deposited on the electrode. In other embodiments, theelectrodes are formed from the high melting point material.

During the current rise time, the plasma will accelerate as it moves inthe plasma propagation channel 114 (e.g., the space between theacceleration electrode 6 and the outer electrode 7) toward the liquidmetal funnel system 120. In some implementations, the plasma acceleratesfor a distance of about 20 m and then for another distance of about 20 mor so to finish discharging the capacitor bank 11. In such embodiments,the total length of the plasma accelerator 110 is about 40 m. Differentlengths of the plasma accelerator are possible. For example, the voltageon the capacitors in the bank 11 can be increased while the capacitanceof the bank 11 is reduced, thereby maintaining the energy stored in thecapacitor bank 11. This can reduce the current rise time and length ofthe accelerator 110. Use of higher voltage in some implementations mayhave possible disadvantages such as being technologically challengingand expensive.

In some embodiments, the plasma accelerator 110 comprises a plasmarestrictor 23. The length of an embodiment of the accelerator 110comprising a plasma restrictor can be less than the length of anaccelerator embodiment that does not comprise a plasma restrictor. Inthe embodiment shown in FIG. 1B, the plasma restrictor 23 is disposednear the first end 112 a of the accelerator 110, for example, below thefirst end 112 a. As the current to the electrodes 6 and/or 7 increases,the magnetic field of the plasma accelerator 110 is initiallyinsufficient to force the plasma past the restrictor 23. Movement of theplasma along the propagation channel 114 is inhibited. The system can beconfigured such that as the magnetic field of the accelerator 110increases (e.g., as the current and/or voltage supplied to theelectrodes 6 and/or 7 increases), the magnetic field increases andreaches a threshold value at which the magnetic force is sufficient toforce the plasma past the restrictor 23. The plasma then acceleratesalong the propagation channel 114. For example, the system may beconfigured such that at (or near) peak current the magnetic force issufficient to push the plasma through the restrictor 23 and to start theplasma accelerating down the plasma accelerator 110.

In some embodiments, the plasma restrictor 23 comprises a constrictionin the plasma propagation channel 114. For example, the constriction maycomprise a narrowing of the space between the acceleration electrode 6and the outer electrode 7. In some embodiments, the constriction isprovided by disposing one or more structures 23 a in the plasmaacceleration channel 114 (see, e.g., FIG. 1B). In other embodiments, theconstriction in the plasma channel 114 is provided by shaping the outerelectrode 7 and/or the acceleration electrode 6 so that the channel 114narrows at the location of the constriction. The location, shape, size,spacing, and/or number of constrictions (e.g., structures disposed in oralong the channel 114 or narrowings of the channel 114) disposed alongthe propagation channel 114 can be selected to provide a desired amountof restriction against movement of the plasma along the channel 114. Insome such implementations, an accelerator length of about 10 m can beused to transfer the energy from the capacitor bank 11 to the plasmausing an 88 kV capacitor voltage.

In other embodiments, the plasma restrictor 23 can be configureddifferently than the constriction schematically illustrated in FIG. 1B.For example, FIG. 1C schematically illustrates an embodiment in whichthe plasma restrictor 23 comprises one or more magnetic coils 23 bdisposed near the first end 112 a of the accelerator 110. When a currentis supplied to the one or more magnetic coils 23 b, the coils 23 bprovide a restricting magnetic force than inhibits movement of theplasma toroid past the position of the coils until the magnetic forceproduced by the accelerator 110 is above a threshold. In someembodiments, the current to the coils 23 b can be reduced (or switchedoff) to reduce (or substantially eliminate) the restricting magneticfield in order to permit the plasma toroid to accelerate toward thesecond end 112 b of the accelerator 110. In some embodiments, acombination of the increase in the magnetic force provided by theaccelerator 110 and the reduction of the restrictive magnetic forceprovided by the coils 23 b permits the plasma to accelerate along thechannel 114 at an appropriate time. In some embodiments, the magneticfield produced by the coils 23 b is sufficient to inhibit accelerationof the toroidal plasma along the channel 114, even at maximum current inthe accelerator 110. In such embodiments, the current to the coils 23 bis reduced (or switched off) to release the toroidal plasma at a desiredtime. In some embodiments, the plasma restrictor 23 can comprise one ormore constrictions 23 a and one or more coils 23 b

After the plasma is formed, the plasma electrically contacts the outerelectrode 7 and the acceleration electrode 6. This contact can shortcircuit the acceleration capacitor bank 11 and start current flowingthrough the plasma. As discussed above, it may be desirable in someimplementations to delay the acceleration (e.g., for about 30 μs, insome cases) to allow the closed magnetic surfaces to form and/or forturbulence, if present, to settle. In some such implementations, asaturable inductor 17 (see, e.g., FIG. 1B) is used to delay the voltagesprovided to the electrodes. The saturable inductor 17 can be disposed intransmission line 15. The saturable inductor 17 may comprise a saturablemagnetic material. For example, an amorphous metal such as, e.g.,METGLAS 2605Co (available from Metglas Inc., Conway, S.C.) can be used.To provide a delay of about 30 μs at a voltage of about 88 kV theinductor 17 can store about 88 kV multiplied by 30 μs=2.6 V·s. In someembodiments, the inductor 17 is substantially toroidal with across-section of about 0.6 m² and a major radius of about 1 m. In someembodiments, the inductor 17 comprises Metglas wound tape with asaturation field of about 1.8 Tesla to provide an appropriate delay.

In the embodiments schematically illustrated in FIGS. 1A-1D, the systemis configured so that at or before the point where the pressure of theplasma builds so that it could exceed the break point or strength of thematerial or the assembly comprising the electrodes 6 and/or 7 (or othercomponents of the accelerator near the second end 112 b), the plasmaexits the accelerator 110 and enters the liquid metal funnel system 120.An advantage of some of these embodiments is that the pressure of theplasma in the accelerator is increased to a relatively large valuewithout damaging the accelerator (e.g., due to yield failure and/ordeformation of second end 112 b of the accelerator 110).

In the illustrated embodiments, the liquid metal funnel system 120comprises a tank 10 and one or more pumps 9 configured to circulate theliquid metal to form a liquid metal funnel 8. The liquid metal flowsfrom a top end of the funnel system 120 to a bottom end of the funnelsystem 120 under the influence of gravity. In some embodiments, the topend of the funnel system 120 is substantially above the bottom end ofthe funnel system 120. In some embodiments, the pumps 9 may provide apressure to the liquid metal, which may also influence the flow of theliquid metal in the tank 10. In some implementations, the funnel 8 has asubstantially cylindrical shape having a passage 125 that issubstantially aligned with a longitudinal axis of the plasma accelerator110. The cross-section of the passage 125 (perpendicular to alongitudinal axis of the passage 125) may be substantially circular,substantially oval, substantially polygonal, or any other shape. Thecross-sectional shape (and/or size) of the passage can change from thetop end to the bottom end. For example, the cross-sectional area at thebottom end may be less than the cross-sectional area at the top end. Thepassage 125 may have an inner surface having an inner diameter. Thecross-section of the inner surface can be substantially circular,substantially oval, substantially polygonal (e.g., rectangular), or anysuitable shape. The inner diameter at the bottom end can be less thanthe inner diameter at the top end. The cross-sectional shape, size,and/or the inner diameter of the passage can be configured to provide adesired amount of compression for the plasma as it moves below the topend. For example, in some embodiments, the inner diameter of the passage125 at the bottom end is about a factor of 3 smaller than the innerdiameter of the passage 125 at the top end. The ratio of the innerdiameter of the passage at the top end to the inner diameter of thepassage at the bottom end can be about 1.5, about 2, about 4, about 5,about 10, about 15, or greater. This ratio can be in a range from about1.5 to about 5, from about 2 to about 4, or some other range.

In some embodiments, the plasma may move from the top end to the bottomend of the passage. In other embodiments, the plasma pressure may becomesufficiently large during movement of the plasma along the channel thatthe plasma may disrupt the funnel 8 before the plasma reaches the bottomend of the passage.

In certain embodiments, the liquid material comprising the liquid funnel8 does not substantially rotate around an axis of the passage. In otherembodiments, the liquid material can be introduced into the tank 10 sothat the liquid material rotates around the axis of the passage as theliquid material moves from the top end to the bottom end. Funnels inwhich the liquid material possesses some amount of rotation (orswirling) may provide advantages in some implementations such as, e.g.,increasing stability of the inner surface of the passage.

Because the plasma can move at a speed (e.g., about several tens ofkm/s, or higher, in some cases) that is higher than the speed of soundin the liquid metal (e.g., about 3 km/s, in some cases), the liquidmetal does not have time to move out of the way as the plasma movesthrough the liquid funnel system 120 (e.g., the inertia of the liquidmetal funnel at least partially confines the plasma). The liquid metaltherefore tends to act as if it were a solid to the plasma and can actto confine the plasma in the passage 125 of the funnel 8. The plasma canexperience compression (and heating) in the funnel 8 as the plasma movesfrom the top end of the funnel 8 to the bottom end of the funnel 8. Forexample, a pressure of the plasma when the plasma is below the top endof the funnel 8 can be greater than a pressure of the plasma when theplasma is above the top end of the funnel 8. FIGS. 1A-1C schematicallyillustrate the compressed plasma toroid 13 in the funnel 8.

The radial compression of the plasma in the passage 125 of the funnel 8can be about 3:1 (or greater) in some implementations. In otherimplementations, the radial compression of the plasma can be about1.5:1, about 2:1, about 4:1, about 5:1, about 7:1, about 10:1, about15:1, or greater. The radial compression of the plasma in the passage125 of the funnel 8 may be in a range from about 1.5:1 to about 5:1,from about 2:1 to about 4:1, or some other range. In certainimplementations, a desired total radial compression of the plasmameasured between the first end of the accelerator 110 and the finalposition of the plasma in the funnel 8 (e.g., when the plasma pressurebecomes sufficiently large to disrupt the funnel) can be about 200:1,about 150:1, about 100:1, about 90:1, about 75:1, about 50:1, about30:1, about 20:1, about 10:1, or some other value. The total radialcompression can be in a range from about 50:1 to about 150:1, from about75:1 to about 125:1, about 80:1 to about 100:1, or some other range.

In some implementations, the desired total radial compression of theplasma toroid (e.g., from the first end 112 a of the accelerator 110 tothe final position of the plasma in the funnel 8) can be achieved byconfiguring the system 1000 to have a first compression ratio in theaccelerator 110 and to have a second compression ratio in the funnel 8such that the first compression ratio multiplied by the secondcompression ratio equals the desired total compression ratio. Forexample, to achieve a total compression of about 90:1, the accelerator110 can be configured to provide a first compression ratio of about 30:1and the funnel 8 can be configured to provide a second compression ratioof about 3:1. These ratios are not limitations on the disclosed systemsand methods, and continuing with this example, a total compression ratioof 90:1 can be achieved differently in different implementations of thesystem 1000, e.g., about 45:1 in the accelerator and about 2:1 in thefunnel, about 18:1 in the accelerator and about 5:1 in the funnel, andso forth. In some embodiments, the first compression ratio in theaccelerator 110 is selected so that a pressure of the plasma at thesecond end 112 b of the accelerator is at or below the material strengthor breakpoint of the materials or assemblies of materials at the secondend 112 b of the accelerator 110. In some implementations, theaccelerator 110 can be configured to provide a desired first compressionratio more readily than the liquid funnel can be configured to provide adesired second compression ratio. In some such implementations, it maybe advantageous for the accelerator 110 to provide more compression thanthe funnel 8 (e.g., the first compression ratio is larger than thesecond compression ratio).

The liquid funnel 8 can comprise a suitable liquid metal such as, forexample, molten lead-lithium (PbLi) with about 17% lithium (Li). Otherlithium percentages can be used in other embodiments (e.g., 0%, 5%, 10%,15%, 20%, 25%, etc.). Also, other liquid materials (e.g., other liquidmetals, liquid metal alloys, etc.) can be used in other embodiments. Forexample, in other embodiments, substantially pure liquid lithium and/orenriched liquid lithium can be used. In some embodiments, the liquidmetal comprises one or more lithium isotopes, which can absorb neutronsand produce tritium.

In some implementations of a system in which the plasma comprises afusionable material, the plasma can be compressed to a density and/ortemperature sufficient to initiate at least some thermonuclear reactionsin the fusionable material. The thermonuclear reactions may produceneutrons. Some of the neutrons may be used for neutron analysis if thesystem is configured, e.g., as a neutron source. Some of the neutronsmay be absorbed by, e.g., the liquid metal funnel 8 and their energyconverted to heat in the molten funnel. Some of this heat may beextracted to produce electrical power (e.g., via steam turbines) if thesystem is configured, e.g., as an energy source. During or after passageof the plasma through the funnel 8, the liquid metal funnel 8 generallyis at least partially disturbed and/or destroyed (e.g., the liquid metalsplashes outwards but is contained by a tank 10). The pumps 9 circulateliquid metal into the tank 10 to re-form the liquid metal funnel 8 forsubsequent injections (or shots) of the plasma. Accordingly, embodimentsof the system schematically illustrated in FIGS. 1A-1D may be configuredto act as a pulsed source of neutrons and/or energy as plasma toroidsare repeatedly introduced into the liquid metal funnel 8.

In some embodiments (see, e.g., FIGS. 1B-1D), in addition to the liquidmetal funnel 8, the liquid funnel system 120 comprises a substantiallycentral or axial liquid guide 22. For example, the liquid guide 22 canbe substantially aligned with the longitudinal axis 115 of theaccelerator 110 and/or the axis of the passage 125. The liquid guide 22can be configured to stabilize and/or reduce a tendency for tilting ofthe plasma torus in the funnel system 120. In some embodiments, theliquid guide 22 is supplied by liquid metal from a liquid metal storagetank or reservoir 18. The storage tank 18 is disposed between theacceleration electrodes 6 in some embodiments. In the embodiment shownin FIG. 1B, the liquid guide 22 flows at least partially under gravitytoward the bottom of the tank 10, and a pump 21 can be used torecirculate the liquid metal back into the storage tank 18 for reuse. Insome such embodiments, it is advantageous if the liquid metal used forthe funnel 8 and the liquid metal used for the liquid guide 22 comprisethe same material, because of the likely mixing of the liquid metal ofthe guide 22 and the liquid metal of the funnel 8 in the tank 10. Forexample, the liquid metal for the liquid guide 22 can comprise moltenPbLi.

In some such embodiments, the plasma is compressed between the innersurface of the funnel 8 and the outer surface of the liquid guide 22,which advantageously may provide a larger amount of compression thanembodiments not using a liquid guide 22. The size and/or shape of thefunnel 8 and/or the liquid guide 22 can be configured to provide adesired amount of compression and/or heating for the plasma as theplasma moves below the top of the liquid metal funnel system 120. Theliquid guide 22 may be physically and/or electrically isolated from theliquid funnel 8 (and/or the tank 10) in various embodiments.

An advantage of some embodiments of the system that use a liquid guide22 is that the liquid metal is electrically conductive and acts like asubstantially central or axial electrode. In some such embodiments, theelectrical current from the plasma accelerator 110 can continue toprovide a magnetic (and/or electromagnetic) force that pushes on theplasma to provide further compression. Accordingly, some suchembodiments can provide additional compression compared to certainembodiments not comprising the liquid guide, in which compression in thefunnel 8 is provided primarily by the momentum of the plasma.

In certain embodiments, the liquid guide 22 is electrically isolatedfrom the outer electrode 7, to reduce the likelihood of or avoidshorting the electrical circuit. In certain such embodiments, the liquidguide 22 is not provided continuously into the tank 10. For example, theliquid funnel system 120 may comprise a liquid guide injection systemthat injects the liquid guide 22 into the passage of the funnel 8 atdesired times. For example, the liquid guide injection system cancomprise a pulse valve 20 that can be opened shortly before a plasmashot. The plasma shot can be fired before the lower end of the liquidmetal in the liquid guide 22 contacts the bottom of the tank 10 (orcontacts liquid metal at the bottom of the tank 10), because suchcontact is likely to complete the electrical circuit. After each shot,the pump 21 (e.g., an intermittent pump) recirculates some of the liquidmetal in the tank 10 to the storage tank 18. In some embodiments, thepump 21 operates to refill the storage tank 18 when the accelerationelectrode 6 is not at a high voltage (e.g., between shots of plasma). Insome embodiments, a portion of the recirculation plumbing (e.g., areturn pipe 31) used for recirculating the liquid metal into the tank 18comprises an electrically insulated section 19 (see, e.g., FIGS. 1B and1C). The insulator section 19 can be oriented substantially verticallyto permit drainage of residual liquid metal after refilling of thestorage tank 18 to reduce the likelihood of short circuiting theacceleration electrode 6. In some embodiments, the space above the fluidin the storage tank 18 is pressurized (e.g., at a pressure of about 30psi, in some cases) (e.g., about 2.07×10⁵ Pa) with an inert gas such as,e.g., argon. The pressurized inert gas provides a downward force on theliquid in the storage tank 18 that (in combination with gravity) allowsthe liquid metal to be ejected at a desired speed.

Example Magnetized Target Fusion Applications

The following discussion is intended to give illustrative, non-limitingexamples of certain parameters of an embodiment of a system that couldbe used to achieve certain plasma compression values. Variousassumptions are discussed in the context of these examples, and variousequations and example calculations are provided herein to highlight someof the factors and considerations involved in an example embodiment of asystem for compressing a plasma. The following discussion is notintended to limit the scope of the systems and methods described herein,nor end-uses or applications of the disclosed systems. In otherimplementations of the systems and methods described herein, otherequations, parameters, factors, and considerations may be applicable.

Magnetized Target Fusion (MTF) systems typically use significant energy(e.g., about 100 MJ in some cases) to compress the plasma. Forgenerating fusion energy in many systems, the well-known Lawson criteriaindicates that a plasma of density n, maintained at a temperature of 10keV, for a time τ, should be selected so that nτ>10²⁰ m⁻³ s for fusionheating to exceed plasma heat losses. However, the plasma cools downwith a time τ=r²/χ where r is the smallest distance between the hotplasma core and the cold edge of the reactor, and χ is the diffusivity.Therefore, a larger plasma (e.g., larger r) may be beneficial but usesmore energy for its formation, and generally therefore a larger and moreexpensive apparatus.

The energy in the plasma is 3/2 NkT_(i) for the ions and 3/2N kT_(e) forthe electrons, where T_(i) is ion temperature, T_(e) is electrontemperature and N is the number of ions or electrons. The number of ionsand electrons is equal in the case of overall charge neutrality.Assuming T_(i) and T_(e) are the same temperature, then the thermalenergy (E_(th)) in the plasma is 3NkT.

Thus, the following equations are applicable for providing estimates forexample parameters in certain embodiments of the system:nτ>10²⁰ m⁻³ sτ=r ²/χE _(th)=3VnkTwith N=nV, where V is plasma volume, and E_(mag)=E_(th)/β where β is theratio of plasma pressure/magnetic pressure. The total energy is thethermal energy E_(th) plus the magnetic energy E_(mag). For a torus, thevolume is 2π²r²r²R where R is the major radius (around the torus) and ris the minor radius. For a compact torus R is approximately equal to rso the volume can be approximated as 2π²r³.

Combining these equations it is found that the minimal energy to reachthe Lawson criterion at 10 keV temperature in some system embodiments isabout:E=7×10¹⁶(1+1/β)χ^(3/2) n ^(−1/2) Joules with n in m⁻³ and χ in m²/s.

The energy E decreases with increasing density and decreases withdecreasing χ. Diffusion and the value of χ in these systems is a subjectof much research. The value of diffusion in some systems is much largerthan a so-called classical calculation because of complex turbulence.Classical estimates for the value of diffusion generally provide thebest possible diffusion. Many experiments observe a diffusion muchlarger than classical, but less than the so-called Bohm diffusion where:χ_(Bohm)=ρ_(i) v _(i)/16where ρ_(i) is the ion gyroradius and v_(i) is the ion thermal velocity.

Assuming Bohm diffusion (as a worst case example scenario), the minimalplasma energy to achieve the Lawson criterion for various plasmadensities (e.g., at 10 keV and β=0.1 typical of certain spheromaks) canbe predicted from the above equations for the above example system andis shown in the graph illustrated in FIG. 2 (solid line with diamonds).The magnetic pressure of a plasma at a temperature of 10 keV, β=0.1 atvarious densities is also plotted on the graph shown in FIG. 2 (solidline with squares). The maximum pressure that solid material can takebefore breaking is typically about 1×10⁴ atm (e.g., about 1000 MPa). Atthat pressure, the example calculations shown in FIG. 2 indicate theexample system should provide about 100 MJ in the plasma to achievebreak-even, and possibly a few times more than that energy for apractical gain. Assuming a transfer efficiency from the power source tothe plasma of about 50%, the system should provide at least about 200 MJof energy to heat the plasma to a fusion temperature.

Embodiments of the disclosed system configured as an energy source mayprovide advantages. For example, using a liquid metal funnel can allowpressures in the plasma to be achieved that are above the breaking pointof solid materials. Therefore, embodiments of the disclosed systems mayprovide increased plasma density, which advantageously reduces theenergy used by the system. This may also reduce the cost and/or size ofthe system.

In some embodiments of the present systems and methods, the plasmapressure increases as the plasma is accelerated and then compressed asthe plasma moves down the accelerator 110 (e.g., along the propagationchannel 114 between the coaxial tapered electrodes 6, 7). At or beforethe point along the plasma path where the plasma pressure meets and/orexceeds the strength of the confining electrode material, the plasma isdirected into the liquid metal funnel system 120 in which furthercompression occurs. For example, the plasma compression can be about afactor of 30 in the accelerator and about a factor of 3 in the funnelsystem. In some embodiments, the plasma can be accelerated to a speedgreater than about 100 km/s down the accelerator 110. The speed of soundin the liquid metal is generally of the order 3 km/s, so the liquidmetal does not have time to move out of the way, and a high plasmapressure is maintained in the funnel 8. In some implementations, a shockwave wake may be generated in the liquid metal. The energy in the shockwave wake is drawn from the plasma kinetic energy; which can be a newenergy loss mechanism in some such embodiments.

FIG. 3 is a schematic cross-sectional diagram showing an example of atoroidal plasma 30 moving within a portion of a tapered liquid metalfunnel 25. Assuming a plasma speed v_(p) and plasma length L, and aliquid metal with a sound speed of c_(s), the schematic diagram in FIG.3 shows a shock wave trailing behind the plasma. The thickness of theshock wave is about c_(s)L/v_(p). In a time of about L/v_(p) (e.g., thetime for the plasma to travel its own length), the volume of thecompressed liquid in the funnel is about 2πRLc_(s)·L/v_(p). Dividing thecompressed volume by the time gives an estimate for the rate at whichliquid metal is compressed:dV/dt=2πRLc _(s) m³/swhere R is the radius of the liquid metal funnel 25.

As an example, a simple approximate formula for the equation of statefor the liquid metal can be used:P=K(V ₀ /V−1)where K is the volume compression modulus and V₀ is the initial volumeat zero compression.

Accordingly, the compression work, PdV, can be integrated to calculatethe energy stored in the compressed liquid metal in this example:E/V=K[ln(P/K+1)−1/(1+K/P)]J/m ³

The power dissipated in the wake is given by the following formula inthis example:Power=2πRLc _(s) K[ln(P/K+1)−1/(1+K/P)] Watts

In some implementation, electrical currents may be induced in the liquidmetal by the magnetic field of the spheromak. Resistive losses in theliquid metal may reduce the energy in the magnetic field that containsthe plasma, representing another possible energy loss mechanism in somecases. The following illustrative example is used to provide an estimatefor this energy loss mechanism.

The current I flowing in the liquid metal to support the magnetic fieldis:I=LB/μ ₀where L is the length of the plasma, B is the magnetic field in thespheromak (or other suitable compact torus) and μ₀ is the vacuumpermeability.

The thickness t of the sheet of electric current flowing in the metal isgiven by:t=(ητ/μ₀)^(1/2)where η is the electrical resistivity of the metal and τ is the timeduring which the magnetic field is applied to the metal and whereτ=L/v_(p)

The resistance is as follows:Resistance=η2πR/Lt

Thus, the power dissipated Ohmically in the liquid metal is:Power=Resistance I ²=2πRB ²(ηLv _(p))^(1/2)μ₀ ^(−3/2) Watts

In some implementations, there also may be power losses due to turbulenttransport. An estimate for such power losses, using the Bohm diffusionformula, is:Power_(Bohm) =E _(th)/τ_(Bohm)

Bremsstralung radiation losses may occur in some cases and an estimatefor such losses is given by:Power_(Bremsstralung)=1.67×10⁻³⁸ n ² T ^(1/2) Z _(eff) W/m ³where T is in eV and n is in m⁻³ and Z_(eff)=ΣZ²n_(z)/n where Z is theatomic number of the impurity and n_(z) is its density. TheBremsstralung radiation power losses are a function of the square of theimpurity atomic number Z, so having a low impurity content can beadvantageous in some cases, especially for impurities with high atomicnumbers.

Continuing with this illustrative, non-limiting example calculation,dividing the energy in the plasma configuration by these various powerlosses gives a total plasma confinement time τ. Using that confinementtime it is possible to calculate the minimum plasma energy to achieve aLawson break-even condition at various densities for this exampleembodiment of the system. The energy for this embodiment is shown in thegraph of FIG. 4.

Note that the energy used may be more than indicated by the examplegraph in FIG. 2 in some implementations, because energy loss mechanismsparticular to the above example embodiment have been taken into account.The graph in FIG. 4 shows that an example estimate for the minimumplasma energy used is about 3 MJ at an example density of 10¹⁹ cm⁻³.Compare this to the example results shown in FIG. 2 that indicate lessthan about 1 MJ at this example density. The plasma outer radius R is2.4 cm in this example calculation. The confinement time is 10 μs inthis example calculation. The magnetic field is 200 Tesla, and thepressure is 0.16 Mbar, in this example calculation. A possible value forthe speed of the plasma during maximum compression is just above thespeed of sound, for example, about 5 km/s. Based on these examplevalues, the plasma moves only about 50 mm during the time where theplasma temperature and pressure conditions could allow fusion to occur.

As discussed above, during and/or after the passage of the plasma, theliquid metal funnel may tend to be outwardly disrupted in someimplementations of the system. In some such implementations, the systemcan be configured so that the liquid metal funnel will reform after atime L_(fe)/v_(f) where L_(fe) is the length of the funnel and v_(f) isthe speed at which the liquid is expelled from the nozzles (that inputliquid metal into the tank 10 of the funnel system 120). Theseparameters can be used to determine an example estimate for the maximumpulse repetition rate in such an embodiment. In this illustrative,non-limiting example, about 1 m of the liquid metal is used to absorbmost of the neutrons so, for example, a 2 m long liquid metal funnel,where the plasma temperature and pressure conditions are suitable forfusion to occur in the center, would be appropriate for someembodiments. Assuming v_(f) is approximately 10 m/s, the repetition ratecould be about 5 Hz in this example. Finally if the net energy out is ofthe order of the energy in, the plasma will produce approximately 3 MJat 5 Hz yielding a power output of approximately 15 MW, which issuitable for a small power plant. Note that these estimates provide onepossible estimate for the size of a power plant producing break-even inthis example, and a larger plant may provide more power but may costmore to develop and build.

Continuing with this illustrative, non-limiting example, working fromthe conditions at maximum compression and assuming that some plasma guns100 typically produce plasma densities not much in excess of about 10¹⁴cm⁻³, the initial plasma formation at the first end 112 a of the plasmaaccelerator 110 would be about 2.2 m in diameter in order to provide thefinal 2.4 cm radius compressed plasma with a density of about 10¹⁹ cm⁻³.The length of the plasma formation initially will be about 1 m, so thisis about the length estimated for the formation region in this exampleimplementation. Therefore, the ratio of the radial size of the compacttoroid at the first end 112 a of the accelerator 110 to the radial sizeof the compact toroid when the toroid is in the liquid metal funnel 8 isabout 100 to 1 in this example. In other embodiments, this ratio can bedifferent such as, for example, about 5:1, about 10:1, about 25:1, about50:1, about 90:1, about 125:1, about 150:1, about 200:1, or some othervalue.

Assuming, for example, about 33% efficiency of energy transfer betweenthe capacitors and the plasma, about 10 MJ will be used in this example.Typical fast discharge foil capacitors have an energy density of about 1J/cm³, so about 10 m³ of capacitor volume is used in this example.Assuming the capacitors are 1 m high and are packed on both sides of adisk-shaped transmission line 15 associated with the plasma accelerator,a disk of about 2.2 m inside diameter and 2.6 m outside diameter is usedin this example. In some embodiments, this disk transmission line plusinternal inductance of the capacitors have an inductance of about 20 nH.The inductance of the plasma accelerator is about 130 nH in someimplementations. Generally, the higher the voltage in the capacitor, thefaster the discharge. Assuming a voltage of about 88 kV, the capacitorbank can have a capacitance of about 2.6 mF. In such an exampleimplementation, the system will have an LC ringing time of about 100 μs.In one example implementation, for reasonable energy transfer to occur,half the ringing time (e.g., about 50 μs) should be approximately equalto the time for the plasma to accelerate down the accelerator. The finalvelocity advantageously can be high enough so the kinetic energy of theplasma is high enough to compress itself in the liquid metal funnel tomaximum compression. Equating the example energy found above to thekinetic energy:3 MJ=mv ²/2

The mass of the plasma is its volume times the density and is about:m=2 milligrams

Accordingly, in this example, a final speed of the plasma is about 1700km/s. In order for the transit time of the plasma to equal half theringing time, an accelerator length of about 40 m can be used in somecases. The length of the accelerator can be advantageously reduced byusing a plasma restrictor at or near the first end 112 a of theaccelerator 110 (see, e.g., the constriction shown in FIG. 1B or themagnetic coils 23 b shown in FIG. 1C). As the current increases, limitedby the inductance, the plasma is unable to pass trough the constriction.In some embodiments, the system is configured such that only at or nearpeak current is the magnetic force strong enough to force the plasmathrough the constriction and rapidly accelerate the plasma. In some suchembodiments, because the plasma starts accelerating only at (or near)peak current, an accelerator that is only about ¼ the length (e.g.,about 10 m, in some cases) can be used. This would provide a reasonable,practical, realistic implementation of a system for compressing plasmaand, in some cases, initiating fusion reactions. Implementations of sucha system may have other uses as well.

Other Example Embodiments and Useful Applications

As discussed above, certain embodiments of the above-described systemsand methods can be used to compress a plasma that comprises a fusionablematerial sufficiently that fusion reactions and useful neutronproduction can occur. For example, the fusionable material may compriseone or more isotopes of light elements such as, e.g., deuterium,tritium, helium-3, lithium-6, lithium-7, etc. Accordingly, certainembodiments of the system may be configured and operated to act asneutron generators or neutron sources. Neutrons so produced have a widerange of practical uses in research and industrial fields. For example,a neutron source can be used for neutron activation analysis (NAA) whichcan provide multi-element analysis of major, minor, trace, and rareelements in a variety of substances (e.g., explosives, drugs, fissilematerials, poisons, etc.) and can be used in a variety of applications(e.g., explosive detection and identification, ecological monitoring ofthe environment and nuclear waste, etc.). Embodiments of the systemconfigured as a neutron source can also be used for materials research(e.g., analyzing the structure, dynamics, composition, and chemicaluniformity of materials), for non-destructive testing of industrialobjects (e.g., via neutron radiography and/or neutron tomography), andfor many other industrial and technological applications. For example,embodiments of the system may be used for nuclear waste remediation andgeneration of medical nucleotides.

Embodiments of the above-described systems and methods for plasmaheating and compression are also suited for applications in the study ofhigh energy density plasma including, for example, applications inastrophysics and nuclear physics.

Recent advances in energy storage (for example, supercapacitors) andhigh-power semiconductor switching have driven down the cost ofhigh-power electrical components. Further developments in electricalpulse power systems and increasing demand for such components for avariety of applications is expected to make an electrically-driven MTFsystem (and/or neutron source) cost competitive with other approaches.In applications where cost is less of a factor (e.g., fusion spacepropulsion where a lower mass payload may be at a premium), embodimentsof such an electrically-driven system may be already appealing comparedto other possible technologies.

In certain implementations of the systems and methods disclosed herein,achieving plasma compression based on an electrical rather than amechanical approach (e.g., certain piston-based systems) can, in somecases, be expected to reduce system maintenance and offer otheradvantages. For example, in some such implementations, the accelerationsystem can be configured with fewer or no moving parts and can be oflower weight. In some embodiments, synchronization issues are simplifiedrelative to certain embodiments of a piston-based system.

While particular elements, embodiments, examples, and applications ofthe present disclosure have been shown and described, it will beunderstood, that the scope of the disclosure is not limited thereto,since modifications can be made by those skilled in the art withoutdeparting from the scope of the present disclosure, particularly inlight of the foregoing teachings. Thus, for example, in any method orprocess disclosed herein, the acts or operations making up themethod/process may be performed in any suitable sequence and are notnecessarily limited to any particular disclosed sequence. Elements andcomponents can be configured or arranged differently, combined, and/oreliminated in various embodiments. Reference throughout this disclosureto “some embodiments,” “an embodiment,” or the like, means that aparticular feature, structure, step, process, or characteristicdescribed in connection with the embodiment is included in at least oneembodiment. Thus, appearances of the phrases “in some embodiments,” “inan embodiment,” or the like, throughout this disclosure are notnecessarily all referring to the same embodiment and may refer to one ormore of the same or different embodiments. Indeed, the novel methods andsystems described herein may be embodied in a variety of other forms;furthermore, various omissions, additions, substitutions, equivalents,rearrangements, and changes in the form of the embodiments describedherein may be made without departing from the spirit of the inventionsdescribed herein.

Various aspects and advantages of the embodiments have been describedwhere appropriate. It is to be understood that not necessarily all suchaspects or advantages may be achieved in accordance with any particularembodiment. Thus, for example, it should be recognized that the variousembodiments may be carried out in a manner that achieves or optimizesone advantage or group of advantages as taught herein withoutnecessarily achieving other aspects or advantages as may be taught orsuggested herein.

Conditional language used herein, such as, among others, “can,” “could,”“might,” “may,” “e.g.,” and the like, unless specifically statedotherwise, or otherwise understood within the context as used, isgenerally intended to convey that certain embodiments include, whileother embodiments do not include, certain features, elements and/orsteps. Thus, such conditional language is not generally intended toimply that features, elements and/or steps are in any way required forone or more embodiments or that one or more embodiments necessarilyinclude logic for deciding, with or without operator input or prompting,whether these features, elements and/or steps are included or are to beperformed in any particular embodiment. No single feature or group offeatures is required for or indispensable to any particular embodiment.The terms “comprising,” “including,” “having,” and the like aresynonymous and are used inclusively, in an open-ended fashion, and donot exclude additional elements, features, acts, operations, and soforth. Also, the term “or” is used in its inclusive sense (and not inits exclusive sense) so that when used, for example, to connect a listof elements, the term “or” means one, some, or all of the elements inthe list.

The example calculations, results, graphs, values, and parameters of theembodiments described herein are intended to illustrate and not to limitthe disclosed embodiments. Other embodiments can be configured and/oroperated differently than the illustrative examples described herein.

What is claimed is:
 1. An apparatus for compressing plasma, theapparatus comprising: a plasma gun configured to generate a compacttoroid of plasma; a plasma accelerator having a first end, a second end,and a longitudinal axis between the first end and the second end, theplasma accelerator configured to receive the compact toroid at the firstend and to accelerate the compact toroid along the longitudinal axistoward the second end; and a liquid funnel system comprising a liquidfunnel and a liquid guide, the liquid funnel having a substantiallycylindrical passage substantially aligned with the longitudinal axis ofthe plasma accelerator, the liquid funnel having an inner surface, theliquid guide extending along a longitudinal axis of the cylindricalpassage, the liquid guide spaced apart from the inner surface of theliquid funnel thereby providing an annular space between the innersurface of the liquid funnel and an outer surface of the liquid guide,the passage having a first inner diameter at a top end of the passageand a second inner diameter at a bottom end of the passage, the secondinner diameter less than the first inner diameter, the liquid funnelsystem configured to receive the compact toroid from the second end ofthe plasma accelerator and to compress the compact toroid as the compacttoroid moves along the annular space from the top end of the passagetoward the bottom end of the passage, wherein the apparatus isconfigured such that a pressure of the compact toroid when below the topend of the passage is greater than a pressure of the compact toroid whenabove the top end of the passage.
 2. The apparatus of claim 1, whereinthe plasma gun has a gun axis that is substantially aligned with thelongitudinal axis of the plasma accelerator.
 3. The apparatus of claim1, wherein the compact toroid comprises a spheromak.
 4. The apparatus ofclaim 1, further comprising a capacitor bank configured to provideelectrical energy to the plasma gun.
 5. The apparatus of claim 4,further comprising a substantially disk-shaped transmission lineconfigured to electrically couple the capacitor bank to the plasma gun.6. The apparatus of claim 1, wherein the plasma gun comprises at leastone magnetic coil.
 7. The apparatus of claim 6, further comprising acooling system configured to cool the at least one magnetic coil.
 8. Theapparatus of claim 1, wherein the plasma accelerator comprises anelectromagnetic rail gun.
 9. The apparatus of claim 1, wherein theplasma accelerator comprises a plasma restrictor located near the firstend of the plasma accelerator, the plasma restrictor configured toinhibit movement of the compact toroid from above the location of therestrictor to below the location of the restrictor until a magneticfield strength of the accelerator exceeds a threshold value, wherein theplasma accelerator comprises a plasma propagation channel, and theplasma restrictor comprises a constriction in the plasma propagationchannel.
 10. The apparatus of claim 1, wherein the plasma acceleratorcomprises a plasma restrictor located near the first end of the plasmaaccelerator, the plasma restrictor configured to inhibit movement of thecompact toroid from above the location of the restrictor to below thelocation of the restrictor until a magnetic field strength of theaccelerator exceeds a threshold value, wherein the plasma restrictorcomprises at least one magnetic coil disposed near the first end of theplasma accelerator.
 11. The apparatus of claim 1, wherein the plasmaaccelerator comprises an inner electrode and an outer electrode, atleast one of the inner electrode and the outer electrode configured witha taper to provide compression to the compact toroid as the toroid movesfrom the first end to the second end.
 12. The apparatus of claim 11,wherein at least one of the inner electrode and the outer electrode iscoated with a material having a melting point greater than a temperatureof the compact toroid when the compact toroid is in the plasmaaccelerator.
 13. The apparatus of claim 1, further comprising acapacitor bank configured to provide electrical energy to the plasmaaccelerator.
 14. The apparatus of claim 13, further comprising asaturable inductor configured to delay the provision of the electricalenergy from the capacitor bank to the plasma accelerator.
 15. Theapparatus of claim 14, wherein the saturable inductor is disposed in adisk-shaped transmission line configured to transmit the electricalenergy from the capacitor bank to the plasma accelerator.
 16. Theapparatus of claim 14, wherein the saturable inductor comprises anamorphous metal.
 17. The apparatus of claim 1, wherein the apparatus isconfigured such that a pressure of the compact toroid at the second endof the plasma accelerator is less than a material strength of the plasmaaccelerator at the second end of the plasma accelerator.
 18. Theapparatus of claim 17, wherein the apparatus is configured such that apressure of the compact toroid when in the liquid funnel is above thematerial strength.
 19. The apparatus of claim 1, wherein the liquidfunnel comprises a liquid metal.
 20. The apparatus of claim 19, whereinthe liquid metal comprises lead-lithium.
 21. The apparatus of claim 19,further comprising a pump system configured to supply the liquid metalto form the liquid funnel of the liquid funnel system.
 22. The apparatusof claim 1, wherein the liquid funnel comprises a liquid material thatflows at least partially under gravity from the top end to the bottomend.
 23. The apparatus of claim 1, wherein the liquid guide iselectrically conductive.
 24. The apparatus of claim 23, wherein theliquid guide is configured to provide a magnetic force on the compacttoroid when the compact toroid is below the top end of the passage. 25.The apparatus of claim 1, further comprising an injection systemconfigured to inject the liquid guide into the liquid funnel systembefore the compact toroid reaches the top end of the liquid funnel. 26.The apparatus of claim 25, wherein the injection system is configured toprovide the liquid guide into the liquid funnel system as a liquidmaterial that flows at least partially under gravity, the liquid guidenot in electrical contact with the liquid funnel when the compact toroidis below the top end of the funnel.
 27. The apparatus of claim 26,further comprising a recirculation system configured to recirculate aportion of liquid material from the liquid funnel system to a reservoirconfigured to store liquid material for the liquid guide.
 28. Theapparatus of claim 27, wherein the recirculation system comprises anintermittent pump configured to recirculate the portion of the liquidmaterial between successive injections of plasma into the liquid funnelsystem.
 29. The apparatus of claim 28, wherein the recirculation systemfurther comprises a return pipe in fluid communication between theliquid funnel system and the reservoir, the return pipe comprising anelectrically insulated section configured to be electrically isolatedfrom the plasma accelerator.
 30. The apparatus of claim 29, wherein theelectrically insulated section is oriented substantially vertically toprovide drainage of liquid material into the reservoir.